On the Theorem of Wan for K-Quasiconformal Hyperbolic Harmonic Self Mappings of the Unit Disk

Abstract

We give a new glance to the theorem of Wan (Theorem 1.1) which is related to the hyperbolic bi-Lipschicity of the K-quasiconformal, K≥1, hyperbolic harmonic mappings of the unit disk D onto itself. Especially, if f is such a mapping and f(0)=0, we obtained that the following double inequality is valid 2|z|/(K+1)≤|f(z)|≤√K|z|, whenever z∈D.